Question

a) Explain that in a one-dimensional system, although a system with an even number of electrons per primitive unit cell is insulator, this does not apply to three-dimensional systems.
b) How the electron state density behaves around the maximum of the valance band and the minimum of the conductivity band of a semiconductor or insulator.
c) Explain why a fully charged band does not contribute to electrical conductivity when an electric field is applied.
d) How does the Bloch wave vector change over time when an external electric field is applied in the effective mass approach?
e) Show that the state density for free electron gas is not dependent on energy in two dimensions. How to relate to energy in one and three dimensions.

Answer 1

Answer(a):

In a real life, we call materials with very large resistivity insulators and materials with very small conductors . this is not a exact definition, because it does not provide a sharp boundary between conductors and insulators.

Consider, the resistivity as a function temparature (T). if resistivity diverges in the T-->0 limit, the system is an insulator, if the resistivity converges to a finite value the system is conductor.

In practoce, one can not reach T=0 so, we go very low T, and check whether the resistivity increases or decreases as T is reduced. if increases exponentially as T goes down,

We say that the system is an insulator.

Generally insulators are completly filled bands 2 per unit cell. This implies that for weekly interactions materials, in witch  electron-electron interactions plays no important role, even number of electrons per unit cell is necessary condition for a band insulators.

Answer (b):

In non metals , the valence band is the highest range of electron energies in which electrons are normally present at absolute zero temprature while the conduction band is the lowest range of vacant electroic states.

Answer(c):

Electrons in the half filled band therfore do contribute to the electrical conductivity of materials, completely filled bands do contain plenty of electrons but do not contribute to the conductivity of material. This is because the electrons can not gain energy since all energy levels are already filled.